Chemistry 6440 / 7440

Basis Sets for

Molecular Orbital Calculations

Resources

• Foresman and Frisch, Exploring Chemistry with Electronic Structure Methods, Chapter 5

• Cramer, Chapter 6

• Jensen, Chapter 5

LCAO Approximation

• numerical solutions for the Hartree-Fock orbitals only practical for atoms and diatomics

• diatomic orbitals resemble linear combinations of atomic orbitals

• e.g. sigma bond in H2

1sA + 1sB

• for polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)

c

Basis Functions

’s are called basis functions• usually centered on atoms• can be more general and more flexible than

atomic orbitals• larger number of well chosen basis functions

yields more accurate approximations to the molecular orbitals

c

Slater-type Functions

• exact for hydrogen atom• used for atomic calculations• right asymptotic form• correct nuclear cusp condition• 3 and 4 center two electron integrals cannot be

done analytically

)2/exp(32/)(

)2/exp(96/)(

)exp(/)(

2

2/1522

2

2/1522

1

2/1311

rxr

rrr

rr

pppx

sss

sss

Gaussian-type Functions

• die off too quickly for large r• no cusp at nucleus• all two electron integrals can be done

analytically

)exp(/2048)(

)exp(9/2048)(

)exp(/128)(

)exp(/2)(

24/137

224/137

24/135

24/13

rxyrg

rxrg

rxrg

rrg

xy

xx

x

s

Comparison of Slater and Gaussian Basis Functions

Contracted Gaussian Basis Functions

• a fixed linear combination of gaussians to form a more suitable basis function

),()( rgdr sss

s

Basis Sets

• a basis set is set of exponents and contraction coefficients for a range of atoms

• types of basis sets– minimal– double zeta / triple zeta / etc.– split valence– polarization functions– diffuse functions

Minimal Basis Set• only those shells of orbitals needed for a

neutral atom

• e.g. 1s, 2s, 2px, 2py, 2pz for carbon

• STO-3G– 3 gaussians fitted to a Slater-type orbital (STO)– STO exponents obtained from atomic

calculations, adjusted for a representative set of molecules

• also known as single zeta basis set (zeta, , is the exponent used in Slater-type orbitals)

Double Zeta Basis Set (DZ)• each function in a minimal basis set is doubled• one set is tighter (closer to the nucleus, larger

exponents), the other set is looser (further from the nucleus, smaller exponents)

• allows for radial (in/out) flexibility in describing the electron cloud

• if the atom is slightly positive, the density will be somewhat contracted

• if the atom is slightly negative, the density will be somewhat expanded

A double zeta basis set allows for flexibility in the radial size

Split Valence Basis Set

• only the valence part of the basis set is doubled (fewer basis functions means less work and faster calculations

• core orbitals are represented by a minimal basis, since they are nearly the same in atoms an molecules

• 3-21G (3 gaussians for 1s, 2 gaussians for the inner 2s,2p, 1 gaussian for the outer 2s,2p)

• 6-31G (6 gaussians for 1s, 3 gaussians for the inner 2s,2p, 1 gaussian for the outer 2s,2p)

Polarization Functions• higher angular momentum functions added to a basis

set to allow for angular flexibility• e.g. p functions on hydrogen, d functions on carbon• large basis Hartree Fock calculations without

polarization functions predict NH3 to be flat • without polarization functions the strain energy of

cyclopropane is too large• 6-31G(d) (also known as 6-31G*) – d functions on

heavy atoms • 6-31G(d,p) (also known as 6-31G**) – p functions on

hydrogen as well as d functions on heavy atoms• DZP – DZ with polarization functions

Effect of Polarization Functions

Diffuse Functions

• functions with very small exponents added to a basis set

• needed for anions, very electronegative atoms, calculating electron affinities and gas phase acidities

• 6-31+G – one set of diffuse s and p functions on heavy atoms

• 6-31++G – a diffuse s function on hydrogen as well as one set of diffuse s and p functions on heavy atoms

Correlation-Consistent Basis Functions

• a family of basis sets of increasing size • can be used to extrapolate to the basis set limit• cc-pVDZ – DZ with d’s on heavy atoms, p’s on H• cc-pVTZ – triple split valence, with 2 sets of d’s

and one set of f’s on heavy atoms, 2 sets of p’s and 1 set of d’s on hydrogen

• cc-pVQZ, cc-pV5Z, cc-pV6Z• can also be augmented with diffuse functions

(aug cc-pVXZ)

Pseudopotentials, Effective Core Potentials

• core orbitals do not change much during chemical interactions

• valence orbitals feel the electrostatic potential of the nuclei and of the core electrons

• can construct a pseudopotential to replace the electrostatic potential of the nuclei and of the core electrons

• reduces the size of the basis set needed to represent the atom (but introduces additional approximations)

• for heavy elements, pseudopotentials can also include of relativistic effects that otherwise would be costly to treat